Problem: When $\sqrt[4]{2^7\cdot3^3}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers.  What is $a+b$?
Answer: We have  \[\sqrt[4]{2^7\cdot3^3} = \sqrt[4]{2^4}\sqrt[4]{2^3\cdot3^3} = 2\sqrt[4]{8\cdot27} = 2\sqrt[4]{216}.\] Therefore, $a+b = \boxed{218}$.